19 | Miller Planes#

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In materials chemistry, it becomes necessary to describe planes within a crystal structure for two main reasons. (1) The first is to describe the orientation of atoms within a crystal, which in turn determines, for example, why viewing a dichroic crystal like ruby along one direction results in a different color than when viewing it along another. This is also relevant in catalysis, where certain crystal faces are more reactive than others, or in understanding why, under certain conditions, crystals may grow as cubes while under other conditions, they may grow as octahedra. (2) The second reason, as we will see later, is that we can use the idea of planes within a crystal structure containing certain atoms to describe the diffraction of X-rays as they pass through a crystal. This is the basis of X-ray crystallography, which is used to determine the atomic structure of materials and is the primary means by which the 3D atomic structures of all matter have been determined—from simple ionic solids like NiO to very large, complex biological molecules like proteins and DNA.

Miller Indices in 2D#

In 2D, a line can be defined by two points in the plane. Miller indices describe the line using two special points—the intercepts of the line with the basis vectors of the crystal system. If a line intercepts the \(x\) and \(y\) axes at the points (1 0) and (0 1), then the intercepts \((x_{intercept},\,y_{intercept}) = (1,\,1)\), or more specifically for any crystal system, the intercepts would be \((\frac{\vec{a}}{1},\,\frac{\vec{b}}{1})\). Miller indices are never defined for intercepts greater than the length of a basis vector. By symmetry of translation, we can always define an equivalent line within one unit cell of the origin. The Miller index (h, k) for the line is based on the reciprocal of the intercepts, such that if the intercepts are \((\frac{\vec{a}}{h},\,\frac{\vec{b}}{k})\), the Miller index is \((h,\,k)\). A few examples are shown in Fig. 55.

../../_images/miller-indices-intercepts.svg — Fig. 55 The construction of lines in 2D using Miller indices. The Miller indices are the reciprocals of the intercepts for the line with the basis vectors of the crystal system.#

A Miller index describes an infinite set of parallel lines in a 2D crystal system. These parallel lines have intercepts with the unit cell axes at integer multiples of the intercepts described by the Miller index, as shown in Fig. 56. For example, the Miller index (4 3) describes a line with intercepts \(x = \frac{\vec{a}}{4}\) and \(y = \frac{\vec{b}}{3}\), as well as parallel lines with intercepts \(x = \frac{2\vec{a}}{4}\) and \(y = \frac{2\vec{b}}{3}\), \(x = 0\) and \(y = 0\), and \(x = \frac{-\vec{a}}{4}\) and \(y = \frac{-\vec{b}}{3}\), and so on.

../../_images/set-of-miller-planes.svg — Fig. 56 Part of the infinite set of 2D Miller planes described by the Miller index (4,3), each with intercepts that are integer multiples of \(x = \frac{n\vec{a}}{4}\) and \(y = \frac{n\vec{b}}{3}\).#

Just as all lattices are centrosymmetric, so too are all Miller indices. The Miller index (4 3) describes the same set of planes as the Miller index (-4 -3). Of course, their construction within the coordinate system places the lines at two different locations, but the total set of planes described by the Miller indices is the same. In contrast, negating only one of the indices—say, (4 -3)—describes a different set of planes that are a mirror image of the set described by (4 3), as shown in Fig. 57. We will see, when using Miller indices to describe the faces of a bulk crystal, that face indices are distinguishable from their inverses, where they assign parallel faces on opposite sides of the crystal.

../../_images/symmetry-miller-planes.svg — Fig. 57 Relationships between Miller indices and their multiplicative inverses.#

Miller Indices in 3D#

Now, considering three dimensions, the construction of Miller planes is the same, except now three points (i.e., three intercepts) are required to define a plane. The intercepts occur at \(x = \frac{\vec{a}}{h}\), \(y = \frac{\vec{b}}{k}\), and \(z = \frac{\vec{c}}{l}\), and the Miller index is denoted as \((h,k,l)\). Shown in Fig. 58 are four examples.

Perhaps the easiest way to assign an index to a set of planes is to count the number of steps it takes for the set to traverse the unit cell along each axis. In the first example shown below, the Miller planes are parallel to \(\vec{a}\) and \(\vec{b}\), so there is no intersection, meaning \(h=0\) and \(k=0\). The planes do intersect the \(c\)-axis at \(\frac{\vec{c}}{1}\), so \(l=1\). We must take one step from the origin to the next plane to traverse the unit cell along \(c\), therefore \(l=1\). In the second example, we must instead take four steps along \(c\) to traverse the unit cell, so \(l=4\), and the Miller index for this set of planes is \((0\,0\,4)\).

In the third example above, all the planes are parallel to the \(c\)-axis (\(l=0\)). There are intersections along the \(a\) and \(b\) axes. Going from plane to plane, we must take one step along the \(\vec{a}\) axis to traverse the unit cell (\(h=1\)) and one step along \(\vec{b}\) to traverse the unit cell (\(k=1\)). The Miller index for this set of planes is therefore \((1\,1\,0)\). In the last example, we must take two steps along the \(\vec{a}\) and \(\vec{b}\) axes to traverse the unit cell, so the Miller index is \((2\,2\,0)\).

Two more examples are shown in Fig. 59. Note that in the case of \((3\,3\,3)\), while there are 10 planes drawn, the Miller index is determined only by the number of steps it takes to traverse the unit cell along each axis, which is three times along \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), respectively. Also, note that all of the planes drawn in the \((1\,1\,1)\) example are also in the set of planes described by the Miller index \((3\,3\,3)\). These two sets of Miller planes are parallel to each other but are distinguished by the larger distance between adjacent planes in the \((1\,1\,1)\) set compared to the \((3\,3\,3)\) set.

Distances Between Miller Planes (d-spacing)#

Each Miller index describes a set of planes that are both parallel to each other and equidistant from each other. The distance between planes is dependent upon the Miller index and the lattice parameters of the crystal. Since crystal systems are not Cartesian, the formula for calculating distances between planes is not simply based on the Pythagorean theorem. For orthorhombic crystal systems, the distance between planes or d-space, \(d\), is given by the formula:

\[ \frac{1}{d^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2} \;\;\;\text{(orthorhombic)} \]

This formula also works for tetragonal systems where \(a=b\), and \(\frac{1}{d^2} = \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2}\). For cubic systems, the formula is further simplified to \(a=b=c\), and the formula becomes \(\frac{1}{d^2} = \frac{h^2 + k^2 + l^2}{a^2}\).

For the other crystal systems, simplifications can also be made, but generally for any crystal system, given the Miller index \((h\,k\,l)\) and the lattice parameters \(a,\,b,\,c,\,\alpha,\,\beta,\,\gamma\), the distance between planes can be calculated as:

\[ d^{*2} = h^2a^{*2} + k^2b^{*2} + l^2c^{*2} + 2hka^*b^*\cos(\gamma^*) + 2hla^*c^*\cos(\beta^*) + 2klb^*c^*\cos(\alpha^*) \]

where \(\vec{a}^* = \frac{1}{V_{cell}} \vec{b} \times \vec{c}\), \(\vec{b}^* = \frac{1}{V_{cell}} \vec{c} \times \vec{a}\), and \(\vec{c}^* = \frac{1}{V_{cell}} \vec{a} \times \vec{b}\) are the reciprocal lattice vectors, \(V_{cell}\) is the volume of the unit cell, and \(\alpha^*,\,\beta^*,\,\gamma^*\) are the respective angles between the reciprocal lattice vectors. Conceptualizing crystals in terms of reciprocal space is a powerful tool for understanding X-ray diffraction and chemical bonding in materials, which we will return to in later units.

Crystal Shapes, their Faces, and Zones#

The natural growth of a crystal in an ideal isotropic environment will result in a crystal with the same point symmetry and crystal system as that of the atomic crystal structure. Crystal growth occurs along certain Miller planes and based on the relative angles between plane the Miller index a given face can be determined. Below in Fig. 60 is shown one example of the shape of a bulk crystal, say a cubic NaCl crystal which most often has the habit of growing large cube-shaped crystals. By imagining a unit cell with its origin at the center of the bulk crystal, the faces of the crystal can be assign Miller indices by considering their intercepts with the basis vectors of the unit cell exactly as is done for constructing Miller planes. within the unit cell. In doing so, the exact orientation of atoms within a large single crystal can be determined as well as specifically how atomic structure are oriented on the surface of the crystal. In heterogeneous catalysis for example, it have been found that certain crystal faces are more reactive than others and this is directly related to the differences in atomic structure along those crystal faces.

Zones#

Just like any other object within a crystal with a defined space and point symmetry, Miller planes can also be grouped into sets of symmetry-equivalent planes. By considering only symmetry operations of pure rotation (i.e., the point group of the crystal), a set of Miller planes can be grouped into a zone. For example, when NaCl forms crystals with a cubic habit, the six faces of the crystal with the Miller indices (1 0 0), (0 1 0), (0 0 1), (-1 0 0), (0 -1 0), and (0 0 -1) are all symmetry equivalent within the point group of the crystal \(m\bar{3}m\) and together compose the zone {1 0 0}. This is because the crystal can be rotated by 90 degrees about the [1 1 1] axis and related to their negated indices by inversion.

Below are a few examples of different zones within a cubic crystal with \(Fm\bar{3}m\) symmetry (the same as rock salt). Some zones have more equivalent planes than others. For example, the zone {1 0 0} has 6 equivalent planes (forming a cube), while the zone {1 1 0} has 12 equivalent planes (forming a dodecahedron), and the zone {1 1 1} has 8 equivalent planes (forming an octahedron). As crystals grow naturally, they may take on shapes composed of multiple zones. The last example in Fig. 60 shows a crystal where one set of faces can be assigned to the zone {1 0 0}, and the remaining 8 faces can be assigned to the zone {1 1 1}.

Depending on the crystal system and the point symmetry of the crystal, the sets of faces that form a zone are different. Below in Fig. 61 are two examples: one is a cubic crystal with \(P23\) symmetry showing a crystal shape formed from the zone {1 1 1}, and the other is a tetragonal crystal with \(P4/mmm\) symmetry showing a crystal shape formed from two zones: {1 1 0} (the 4 sides) and {1 1 1} (the 8 faces forming the pyramidal caps).

Note that in the case of the tetragonal system, the zone {1 1 0} on its own cannot describe all of the faces of a crystal as was the case for the cubic system. This is because the zone {1 1 0} does not contain any faces that intersect the \(c\)-axis. Another zone, like {1 1 1} (pyramidal caps, an elongated square bipyramidal habit) or {0 0 1} (flat caps or a square prismatic crystal habit), is needed to describe a complete 3D polyhedron in the tetragonal system. In contrast, the zone {1 1 1} alone in the tetragonal system could describe a crystal shape on its own, since it contains faces that intersect all three unit cell axes to form a closed polyhedron. A tetragonal crystal with only {1 1 1} faces would have a square bipyramidal crystal habit (i.e., an elongated octahedron).

Terminology#

Miller Plane#: A plane within a crystal structure that is described by its intercepts with the basis vectors of the unit cell. The Miller index of a plane is denoted as \((h\,k\,l)\), where \(h,\,k,\,l\) are the denominators of the intercepts of the plane with the basis vectors, expressed as \(\text{intercepts}=(\frac{\vec{a}}{h},\frac{\vec{b}}{k},\frac{\vec{c}}{l})\).
Crystal Habit#: The shape of a crystal that results from the natural growth of a crystal. The crystal habit is determined by the relative growth rates of the crystal faces and the symmetry of the crystal.
Crystal Face#: A 2D plane corresponding to a face of a grown single crystal that can be indexed using Miller indices.
Zone#: A set of symmetry-equivalent Miller planes within a crystal that are related by the point group symmetry of the crystal. Zones are used to group faces of a crystal that are symmetry equivalent. Zones are denoted by the Miller index of a single plane within the zone, contained within curly brackets, e.g., \({1\,1\,1}\).

19 | Miller Planes

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